Fundamental and Singular Solutions
We construct a matrix of fundamental solutions for the operatorA ω (∂ x )using the method described in [13] IfA ω∗ (ξ)is the adjoint of the matrixA ω (ξ), then u(x) =A ω∗ (∂ x )B(x), (2.1) whereBsatisfies
(detA ω )(∂ x )B(x) =H(x) (2.2)From (1.10) and (1.11) it follows that detA ω (ξ)
Factoring this expression, we obtain detA ω (ξ) =h 4 μ 2 (λ+2μ) Δ 2 +λ+3μ λ+2μ k 2 Δ+ μ λ+2μ k 2 k 2 3
(Δ+k 2 3 ), where k 2 =ρω 2 μ (2.3) and k 2 3 =k 2 − 1 h 2 ; (2.4) hence, detA ω (ξ) =h 4 μ 2 (λ+2μ)(Δ+k 2 1 )(Δ+k 2 2 )(Δ+k 2 3 ), (2.5) where k 2 1 +k 2 2 =λ+3μ λ+2μ k 2 , k 2 1 k 2 2 = μ λ+2μ k 2 k 3 2
Without loss of generality, we assume thatk 2 1 ≥k 2 2
Using (2.4) and (2.6), we find thatk 2 1 ,k 2 2 , andk 2 3 are connected by the equality h 2 (k 2 1 −k 2 3 )(k 2 2 −k 2 3 ) =h 2 k 2 1 k 2 2 −k 2 3 (k 2 1+k 2 2 ) +k 4 3
We claim that, under assumptions (1.14) and (1.15),k 2 1 ,k 2 2 , andk 3 2 are real, strictly positive, and distinct First,k 2 1 andk 2 2 are the two roots of the equation x 2 +λ+3μ λ+2μ k 2 x + μ λ+2μ k 2 k 3 2 = 0
The discriminant of this quadratic is
Consequently,k 2 1 andk 2 2 are real and distinct Also, by (2.4), assumption (1.15) implies thatk 2 3 >0 By (2.6) and (1.14), this means that k 2 1 +k 2 2 >0, k 2 1 k 2 2 >0.
Hence,k 2 1 andk 2 2 are strictly positive Finally, from (2.7) and the fact thatk 2 3 =0 it follows thatk 2 1 =k 2 3 andk 2 2 =k 2 3
Replacing, in turn, each component ofHby−δ(|x−y|), whereδ is the Dirac delta distribution, and setting the other two equal to zero, from (2.1) and (2.2) we obtain the matrix of fundamental solutions
, (2.8) where, by (2.2) and (2.5),t(x,y)is a solution of h 4 μ 2 (λ+2μ)(Δ+k 2 1 )(Δ+k 2 2 )(Δ+k 2 3 )t(x,y) =−δ(|x−y|) (2.9)
In our analysis, we express the function t(x,y) as a summation of Hankel functions, specifically t(x,y) = ∑_{j=1}^{3} b_j H_0^{(1)}(k_j |x−y|), where H_0^{(1)} represents the Hankel function of the first kind of order zero The constants b_j are determined from the preceding equation (2.9) Notably, the Hankel function serves as a fundamental solution to the Helmholtz operator, adhering to established properties in the field.
From (2.10) and (2.11) we find that
To eliminate the Dirac distribution in this equation, we require that b 1+b 2+b 3=0 (2.12)
Now, if (2.12) is satisfied, we see that
+4ib 2(k 3 2 −k 2 2 )δ(|x−y|), so we must have
=4ib 1(k 1 2 −k 2 2 )(k 2 1 −k 2 3 )δ(|x−y|), from which we deduce that b 1= i
4h 4 μ 2 (λ+2μ)(k 2 1 −k 2 2 )(k 1 2 −k 2 3 ) (2.14) Substituting this into (2.12) and (2.13) yields b 2= i
The formula forb 3can be simplified by means of (2.7): b 3=− i
These constants are well defined since thek 2 j are distinct.
To calculate the matrix of fundamental solutionsD ω (x,y)using (2.8), we first need to computeA ω∗ (∂ x ) From (1.10) and (1.11) it follows that
Therefore, the elements of the adjoint of the matrixA ω (ξ)are
Theorem 2.1 The elements of the matrix of fundamental solutions D ω (x,y)are
Proof First, we rewrite the elements of the adjoint matrix in a more convenient form Using (2.3) and (2.4), from (2.17) we obtain
−h 2 μ(λ+μ)(Δ+k 2 )ξαξ β −μ 2 ξαξ β , so, using (2.4) and (2.6), we arrive at
Taking (2.14)–(2.16) into account, we find that
Now, using (2.7) and (2.26), we deduce that α1 2 = k 2 3 k 2 2 −k 1 2
Similarly, it can be shown that α2 2 =k 2 1 −μ k 2 3 k 2 1 −k 2 2
, whereβ1 2 andβ2 2 are defined by (2.25) This completes the proof
The representation of D ω (x,y) provided in Theorem 2.1 is adequate for our needs, eliminating the necessity to explicitly compute the first-order and second-order derivatives of the Hankel functions as outlined in equations (2.20)–(2.22).
To simplify the notation, we writer=|x−y| Since
∂ y α∂ y β f (r), from (2.20)–(2.22) it is easily seen that
We introduce the matrix of singular solutions
Theorem 2.2 Each column of D ω (x,y)and P ω (x,y)satisfies the homogeneous sys- tem A ω (∂ x )u(x) =0at all points x∈R 2 , x=y.
Proof From (2.8), (2.5), and (2.9) we see that forx=y,
Using (2.28) and (2.27), we find that forx=y,
=T jm (∂ y )A ω ik (∂ x )D km ω (x,y) =0,which proves the assertion.
Order of Singularity
The investigation of the singularities ofD ω andP ω plays an important role in the study of the behavior of the single-layer and double-layer plate potentials.
It is known [1] that, asξ →0,
64ξ 4 − ããã lnξ; so, asr→0, from (2.10) we see that t(x,y) =t(r) =∑ 3 j=1 b j H 0 (1) (k j r)
64k 4 j r 4 lnr+d+O(r 6 lnr), wheredis a constant From (2.14) and (2.15) it follows that
4h 4 μ 2 (λ+2μ), which leads to t(r) =c 1 r 4 lnr+d+O(r 6 lnr), (2.29) where c 1=− 1
(2.32) andD˜ ω (x,y)is a(3×3)-matrix whose elements are O(rlnr)as r→0.
Proof By (2.8), (2.17), (2.29), and (2.30), we find that
∂ x α∂ x β (c 1 r 4 lnr) +ρω 2 (ρω 2 h 2 −μ)δ αβ c 1 r 4 lnr+C αβ +O(r 2 lnr) dc 1 h 2 μ(λ+2μ)δ αβ ln r −c 1 h 2 μ(λ+μ) ∂ 2
Also, using (2.18) and (2.19), we see that
+ (ρω 2 h 2 −μ) 2 c 1 r 4 lnr+C 33 +O(r 2 lnr) dc 1 h 4 μ(λ+2μ)lnr+C 33+O(r 2 lnr)
We introduce the alternating symbolε αβ =β−αforα,β∈ {1,2}.
∂s(y)denote the derivatives in the normal and tan- gential directions, respectively.
+D+P˜ ω (x,y), (2.33) where D is a constant(3×3)-matrix, λ = λ (λ+2μ), μ is defined by (2.26), andP˜ ω (x,y)is a(3×3)-matrix whose elements are O(rlnr) as r→0.
Remark 2.1.Expansions (2.31) and (2.33) ofD ω (x,y) andP ω (x,y), respectively,for yclose tox coincide with those of the corresponding matrices arising in the equilibrium bending of plates, which can be found in [14].
Properties of the Potentials
We define the single-layer potential
D ω (x,y)ϕ(y)ds(y) (2.34) and the double-layer potential
P ω (x,y)ϕ(y)ds(y), (2.35) whereϕis a(3×1)-vector function known as the density.
The properties of the potentials play a crucial role in developing suitable integral equations for various boundary value problems, particularly focusing on the behavior of (V ω ϕ)(x) and (W ω ϕ)(x) as x approaches ∂S This necessitates a detailed analysis based on specific expansions Notably, the singularities of D ω and P ω align with those found in the corresponding matrices related to the equilibrium bending of plates A comprehensive investigation into the behavior of these potentials is provided in [14], where many relevant properties are referenced.
In the context of functional analysis, we define the vector space \( C^{0,\alpha}(X) \) as the collection of H¨older continuous functions with index \( \alpha \) on the set \( X \) Additionally, \( C^{1,\alpha}(X) \) represents the subspace of \( C^1(X) \) consisting of functions whose first-order derivatives are also in \( C^{0,\alpha}(X) \) When we refer to a function \( \phi \in C^{0,\alpha}(X) \) with \( \alpha \) in the range \( (0,1) \), we imply that \( \phi \) satisfies a specific equation or condition for all values of \( \alpha \) within this interval.
The next three assertions follow from Theorem 2.2 and the results in [14].
Theorem 2.5 (i) If ϕ ∈C(∂ S ), then V ω ϕ and W ω ϕ are analytic and satisfy
(ii) Ifϕ∈C 0,α (∂S), α∈(0,1), then the direct values V 0 ω ϕ and W 0 ω ϕ of V ω ϕ and W ω ϕ on ∂S exist (the latter in the sense of principal value).
Theorem 2.6 Ifϕ∈C 0,α (∂S),α∈(0,1), then the functions
S − (2.36) are of class C ∞ (S + )∩C 1,α (S¯ + )and C ∞ (S − )∩C 1,α (S¯ − ), respectively, and
TV ω− (ϕ) W 0 ω∗ − 1 2 I ϕ on ∂S, (2.38) where W 0 ω∗ is the adjoint of W 0 ω and I is the identity operator.
Theorem 2.7 Ifϕ∈C 1,α (∂S),α∈(0,1), then the functions
W 0 ω + 1 2 I ϕ on ∂ S , (2.40) are of class C ∞ (S + )∩C 1 ,α (S¯ + )and C ∞ (S − )∩C 1 ,α (S¯ − ), respectively, and
Remark 2.2.Theorems 2.5–2.7 are used in Chapters 6–11 to rigorously justify the construction of regular solutions of the fundamental boundary value problems inS + andS − in the form of layer potentials.
Analyzing System (1.10) presents challenges due to the nonhomogeneous term on the right-hand side In this chapter, we develop a specific solution for the system by utilizing a domain potential, which is defined through the kernel of the fundamental solutions matrix for the operator A ω (∂ x) introduced in Chapter 2.
The findings are relevant to the equilibrium bending of thin elastic plates characterized by the system A(∂ x )u(x) = 0, which accounts for transverse shear deformation This connection arises from the analytic argument focusing on the singularities of the fundamental solutions matrix as r approaches zero, aligning with the singularities of the corresponding matrix for operator A(∂ x ) These results are essential for demonstrating the existence of nonzero solutions in homogeneous boundary value problems related to stationary plate oscillations.
This article examines the continuity of the first-order derivatives of the domain potential, as outlined in Theorems 3.1 and 3.2 The subsequent Theorems 3.3 to 3.5 analyze the behavior of the second-order derivatives Ultimately, Theorem 3.6 consolidates these findings, demonstrating that the potential is a sufficiently regular solution to the nonhomogeneous system.
The Newtonian Potential
We intend to construct a particular solution of the system
Here, the matrix operator B(∂ x ) represents either A ω (∂ x ), defined by (1.10), or
S k(x,y)f(y)da(y), (3.1) whereSdenotesS + orS − andk(x,y)is eitherD ω (x,y)or D(x,y), the matrix of fundamental solutions for the operatorA(∂ x )introduced in [14] Such functions are known as Newtonian potentials (see, for example, [27]).
For clarity, we focus on the interior domain S +, while the exterior domain is addressed similarly, requiring additional conditions on f(x) as |x| approaches infinity to guarantee the existence of K(x) as an improper integral.
From Theorem 2.3 and Remark 2.1 it follows that, asr→0, k(x,y) =lnr d 1 E γγ − 1
+C+O(rlnr), (3.2) whereCis a constant(3×3)-matrix andd 1andd 2are defined by (2.32).
In this chapter, we differentiate the series expansion (3.2) to derive the expansions for the derivatives of the fundamental solutions' matrices This process is supported by the inherent properties of the series expansions related to Hankel functions.
In what follows,σ(a,r)is the disk with the center ata, radius r, and circular boundary∂σ(a,r).
Smoothness Properties
Existence of the First-Order Derivatives
The first assertion addresses the differentiability of the Newtonian potentialK.
∂ x α K(x)exists at each point x∈S + (x∈∂ S ) and ∂
It is clear thatk δ (x,y)has first-order derivatives with respect tox∈S + (x∈∂ S). Hence, ∂
∂ x α K δ (x)obviously exists at each pointx∈S + (x∈∂ S) and
∂ x α k δ (x,y)f(y)da(y), where the kernel of the integral satisfies
|k(x,y)|da(y)→0, uniformly asδ →0, sincek(x,y)has only a logarithmic singularity; that is, its be- havior is similar to δ 0 rlnr dr= 1
1 rda(y)→0, uniformly asδ →0 The second integral tends to zero because, asδ→0, it is of the same order as
The assertion now follows from a well-known theorem of real analysis.
H¨older Continuity of the First-Order Derivatives
Before discussing the H¨older continuity of the derivatives ofK, we require a pre- liminary result.
Lemma 3.1 Ifβ∈(0,1], then there is a constant c such that
|x −x |≤c|x −x | α , 0